variance statistics jee mains notes

Sometimes we have to take the mean deviation by taking the absolute values from a set of values. The absolute values were taken to measure the deviations, as otherwise, the positive and negative deviation may cancel out each other.

So, to remove the sign of deviation, we usually take the variance of the data set, i.e., we usually square the deviation values. As squares are always positive, so the variance is always a positive number.

Let us take “n” observations as a₁, a₂, a₃, ….., a_n

Then the variance is denoted by

Properties of Variance

• If the variance comes out to be zero, this means that
   
   
• If the variance is small, 
   
  If the value is greater, 
   
• If each observation is increased by a where a ∈ R, then the variance will remain unchanged.
• If each observation is multiplied by a where a ∈ R, then the variance will be multiplied by a² also.
• But for some data sets, the variance by the formula
   
  does not give the proper values as the range of deviation may vary, and the observations may be more scattered about the mean. So, to overcome this difficulty, we take the mean of the square of the deviations.

So, the variance is given by:

As a result of squaring, the unit of variance is not the same as that of the data sets taken.